A bound on the genus of a curve with Cartier operator of small rank

TL;DR

This paper establishes a new upper bound on the genus of algebraic curves in characteristic p with a Cartier operator of rank 1, refining previous bounds for such curves.

Contribution

It improves the known bound on the genus of curves with Cartier operator of rank 1 from Re's result to a tighter limit, advancing understanding of curve classification in positive characteristic.

Findings

New genus bound: p + p(p-1)/2 for curves with Cartier operator of rank 1

Improved upon previous bounds by Re for specific curve classes

Enhances classification criteria for algebraic curves in characteristic p

Abstract

Ekedahl showed that the genus of a curve in characteristic $p>0$ with zero Cartier operator is bounded by $p(p-1)/2$. We show the bound $p+p(p-1)/2$ in case the rank of the Cartier operator is 1, improving a result of Re.